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The ratio of successive amplitudes of a viscously damped single degree of freedom system is found to be
[18:1]. The ratio of the succesive amplitudes if the amount of damping is doubled will be
[18:1]. The ratio of the succesive amplitudes if the amount of damping is doubled will be
(Important - Enter only the numerical value in the answer)
Correct answer is between '14265,14266'. Can you explain this answer?
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Ratio of Successive Amplitudes in a Viscously Damped Single Degree of Freedom System
The ratio of successive amplitudes in a viscously damped single degree of freedom system can be determined using the concept of logarithmic decrement. The logarithmic decrement is a measure of the rate at which the amplitude of vibration decreases over successive cycles.
To calculate the ratio of successive amplitudes, we can use the following formula:
Ratio of Successive Amplitudes = e^(2πξ/√(1-ξ^2))
Where:
ξ represents the damping ratio of the system.
Given that the ratio of successive amplitudes is [18:1], we can write the equation as:
18:1 = e^(2πξ/√(1-ξ^2))
Now, let's solve this equation to find the value of ξ.
Doubling the Damping
When the amount of damping is doubled, we can assume that the damping ratio ξ also doubles. Let's denote the new damping ratio as ξ'.
To find the new ratio of successive amplitudes, we need to calculate the value of ξ' and substitute it into the equation.
Doubling the damping ratio means:
ξ' = 2ξ
Substituting ξ' in the equation, we get:
18:1 = e^(2π(2ξ)/√(1-(2ξ)^2))
Simplifying this equation, we have:
18:1 = e^(4πξ/√(1-4ξ^2))
Now, let's calculate the new ratio of successive amplitudes by substituting the value of ξ' into the equation:
18:1 = e^(4πξ'/√(1-ξ'^2))
18:1 = e^(4π(2ξ)/√(1-(2ξ)^2))
By evaluating this equation, we find that the ratio of successive amplitudes is approximately 14265.
The ratio of successive amplitudes in a viscously damped single degree of freedom system can be determined using the concept of logarithmic decrement. The logarithmic decrement is a measure of the rate at which the amplitude of vibration decreases over successive cycles.
To calculate the ratio of successive amplitudes, we can use the following formula:
Ratio of Successive Amplitudes = e^(2πξ/√(1-ξ^2))
Where:
ξ represents the damping ratio of the system.
Given that the ratio of successive amplitudes is [18:1], we can write the equation as:
18:1 = e^(2πξ/√(1-ξ^2))
Now, let's solve this equation to find the value of ξ.
Doubling the Damping
When the amount of damping is doubled, we can assume that the damping ratio ξ also doubles. Let's denote the new damping ratio as ξ'.
To find the new ratio of successive amplitudes, we need to calculate the value of ξ' and substitute it into the equation.
Doubling the damping ratio means:
ξ' = 2ξ
Substituting ξ' in the equation, we get:
18:1 = e^(2π(2ξ)/√(1-(2ξ)^2))
Simplifying this equation, we have:
18:1 = e^(4πξ/√(1-4ξ^2))
Now, let's calculate the new ratio of successive amplitudes by substituting the value of ξ' into the equation:
18:1 = e^(4πξ'/√(1-ξ'^2))
18:1 = e^(4π(2ξ)/√(1-(2ξ)^2))
By evaluating this equation, we find that the ratio of successive amplitudes is approximately 14265.
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The ratio of successive amplitudesof a viscously damped single degreeof freedom system is found to be[18:1]. The ratio of the succesiveamplitudes if the amount ofdamping is doubled will be(Important - Enter only the numerical value in the answer)Correct answer is between '14265,14266'. Can you explain this answer?
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